On the Densities of Certain Lattice Packings by Parallelepipeds

Abstract

Given any (non-degenerate) n-dimensional lattice L, let κ(L) denote the supremum of the numbers κ such that there exists a lattice packing Q + L of density κ where Q is some o-symmetric parallelepiped with faces parallel to the coordinate axes. Many efforts have been made to determine or estimate the minimal such density κn taken over all n-dimensional lattices. It is known that \(\lim sup_{n \to \infty} (\kappa_n)^{1/n} 0\). Here we investigate a sequence of lattices Ln which are known to minimize the function κ(L) in dimensions n ≦ 3 and are likely to provide the minima κn = κ(Ln) in certain higher dimensions. We establish the inequality κ(Ln) ≧ n−n/2 which supports the conjecture that lim supn → ∞ (κn)1/(n log n) is positive.